Federer–Morse theorem
{{Short description|On a property of surjective continuous maps between compact metric spaces}}
{{distinguish|Morse–Sard–Federer theorem}}
{{context|date=March 2017}}
In mathematics, the Federer–Morse theorem, introduced by {{harvs|txt|author2-link=Anthony Morse|last2=Morse|author1-link=Herbert Federer|last1=Federer|year=1943}}, states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a Borel subset Z of X such that f restricted to Z is a bijection from Z to Y.Section 4 of {{harvs|txt|last=Parthasarathy|year=1967}}.
Moreover, the inverse of that restriction is a Borel section of f—it is a Borel isomorphism.Page 12 of {{harvs|txt|last=Fabec|year=2000}}
See also
References
{{Reflist}}
- {{Citation | last=Baggett | first=Lawrence W. | title=A Functional Analytical Proof of a Borel Selection Theorem | year=1990 | journal=Journal of Functional Analysis | volume=94 | pages=437–450}}
- {{Cite book|first=Raymond C.|last=Fabec|title=Fundamentals of Infinite Dimensional Representation Theory|year=2000|publisher=CRC Press|isbn=978-1-58488-212-1|url-access=registration|url=https://archive.org/details/makingofearth00jone_0/page/12}}
- {{Citation | last1=Federer | first1=Herbert | last2=Morse | first2=A. P. | title=Some properties of measurable functions | doi=10.1090/S0002-9904-1943-07896-2 | mr=0007916 | year=1943 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=49 | pages=270–277| doi-access=free }}
- {{cite book|last=Parthasarathy|first=K. R.|title=Probability measures on metric spaces|series=Probability and Mathematical Statistics|issue=3|publisher=Academic Press, Inc.|location=New York-London|year=1967}}
Further reading
- L. W. Baggett and Arlan Ramsay, A Functional Analytic Proof of a Selection Lemma, Can. J. Math., vol. XXXII, no 2, 1980, pp. 441–448.
{{Use dmy dates|date=December 2017}}
{{DEFAULTSORT:Federer-Morse theorem}}